Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
÷èñåë íå çàìêíåíà â³äíîñíî ä³ëåííÿ. Ìíîæèíà ö³ëèõ ÷èñåë<br />
º óïîðÿäêîâàíîþ ìíîæèíîþ.<br />
1.2.3. Ðàö³îíàëüí³ ÷èñëà<br />
Ðàö³îíàëüíèé äð³á — öå äð³á ó âèãëÿä³ m/n, äå m ö³ëå<br />
÷èñëî, à n íàòóðàëüíå. ßêùî ÷èñåëüíèê ³ çíàìåííèê —<br />
âçàºìíî ïðîñò³ ÷èñëà, òî äð³á íàçèâàºòüñÿ íåñêîðîòíèì. Äâà<br />
ðàö³îíàëüí³ äðîáè m/n ³ m 1 /n 1 íàçèâàþòü åêâ³âàëåíòíèìè,<br />
ÿêùî m·n 1 =n·m 1 . Ðàö³îíàëüíèì ÷èñëîì íàçèâàºòüñÿ ìíîæèíà<br />
âñ³õ åêâ³âàëåíòíèõ ì³æ ñîáîþ ðàö³îíàëüíèõ äðîá³â. Íà<br />
ìíîæèí³ ðàö³îíàëüíèõ ÷èñåë ââîäÿòüñÿ 4 îïåðàö³¿ (àðèôìåòè÷í³<br />
䳿).<br />
m1 m2<br />
Íåõàé r1 = ; r2<br />
=<br />
n n . Òîä³<br />
1 2<br />
m ⋅ n ± m ⋅n<br />
; 3)<br />
1 2 2 1<br />
1–2) r1 ± r2<br />
=<br />
n1⋅<br />
n2<br />
4)<br />
m ⋅ n<br />
r : r = ( r ≠0)<br />
.<br />
1 2<br />
1 2 2<br />
n1⋅<br />
m2<br />
m ⋅ m<br />
r ⋅ r =<br />
1 2<br />
1 2<br />
n1⋅<br />
n ;<br />
2<br />
Âèñíîâîê. Íà ìíîæèí³ ðàö³îíàëüíèõ ÷èñåë îïåðàö³¿<br />
1)–3) çàìêíåí³. ×åòâåðòà îïåðàö³ÿ íà ìíîæèí³ ðàö³îíàëüíèõ<br />
÷èñåë íå çàâæäè çä³éñíþºòüñÿ. Âîíà ïîòðåáóº äîäàòêîâî¿<br />
óìîâè.<br />
Ìíîæèíà ðàö³îíàëüíèõ ÷èñåë º óïîðÿäêîâàíîþ. Äëÿ öüîãî<br />
òðåáà ââàæàòè, ùî áóäü-ÿêå äîäàòíå ÷èñëî ³ ÷èñëî íóëü<br />
º ÷èñëîì á³ëüøèì, í³æ áóäü-ÿêå â³ä’ºìíå ÷èñëî. Äîäàòí³ ÷èñëà<br />
ïîð³âíþþòüñÿ ì³æ ñîáîþ òàê:<br />
m1 m2<br />
r1 = ≤ r2 = ⇔ mn<br />
1 2<br />
≤nm<br />
1 2<br />
n1 n<br />
.<br />
2<br />
ßêùî æ ÷èñëà r 1 ³ r 2 — â³ä’ºìí³, òî ç äâîõ öèõ ÷èñåë<br />
ââàæàºòüñÿ á³ëüøèì òå ÷èñëî, ÿêå çà àáñîëþòíîþ âåëè÷èíîþ<br />
º ÷èñëîì ìåíøèì.<br />
Çàóâàæåííÿ. Çíàê ⇒ îçíà÷àº, ùî ç ïåðøîãî òâåðäæåííÿ<br />
âèïëèâຠäðóãå, à çíàê ⇔ îçíà÷àº, ùî ç ïåðøîãî<br />
òâåðäæåííÿ âèïëèâຠäðóãå ³ íàâïàêè.<br />
Òåîðåìà 1.2.1 (ïðî ù³ëüí³ñòü ðàö³îíàëüíèõ ÷èñåë).<br />
̳æ äâîìà ð³çíèìè ðàö³îíàëüíèìè ÷èñëàìè ³ñíóº íåñê³í-<br />
÷åííà ìíîæèíà ðàö³îíàëüíèõ ÷èñåë.<br />
Äîâåäåííÿ. Ùîá äîâåñòè öå òâåðäæåííÿ, ìè ïîêàæåìî,<br />
ùî ì³æ áóäü-ÿêèìè ð³çíèìè äâîìà ðàö³îíàëüíèìè ÷èñëàìè<br />
r 1 ³ r 2 º ïðèíàéìí³ îäíå ÷èñëî r 3 .<br />
1<br />
+<br />
2<br />
Íåõàé r1 < r2,<br />
r<br />
3<br />
= r r . Ïîêàæåìî, ùî r<br />
2<br />
1 r1 + r2 ⇒ r2 > = r<br />
3.<br />
2<br />
Òåîðåìó äîâåäåíî.<br />
1.2.4. ²ððàö³îíàëüí³ ÷èñëà<br />
²ððàö³îíàëüí³ ÷èñëà (íåðàö³îíàëüí³ ÷èñëà) âèíèêëè òàêîæ<br />
â ðåçóëüòàò³ ä³ÿëüíîñò³ ëþäåé. ßê ïðèêëàä ìîæíà<br />
âçÿòè çíàõîäæåííÿ äîâæèíè ä³àãîíàë³ êâàäðàòà ç³ ñòîðîíîþ,<br />
ÿêà äîð³âíþº îäèíèö³.  ðåçóëüòàò³ îòðèìàºìî 2 . Ïîêàæåìî,<br />
ùî 2 íå º ðàö³îíàëüíèì ÷èñëîì.<br />
Äîâåäåííÿ öüîãî ôàêòó áóäåìî ïðîâîäèòè ìåòîäîì<br />
â³ä ñóïðîòèâíîãî. Ïðèïóñòèìî, ùî 2 = m / n (ðàö³îíàëüíå<br />
÷èñëî). Ïðè öüîìó áóäåìî ââàæàòè, ùî m/n — íåñêîðîòíèé<br />
äð³á. Òîä³ ìàºìî:<br />
2<br />
m m 2 2<br />
2 = ⇒ 2= ⇒ m = 2n<br />
2<br />
.<br />
n n<br />
Öå îçíà÷àº, ùî m 2 — ïàðíå ÷èñëî. Íåâàæêî ïîêàçàòè, ùî<br />
ÿêùî m 2 ïàðíå, òî m òåæ ïàðíå. Áóäü-ÿêå ïàðíå ÷èñëî<br />
ìîæíà çîáðàçèòè ó âèãëÿä³ m=2m 1 , ï³ñëÿ ÷îãî îòðèìàºìî<br />
4m = 2n ⇒ 2m = n . Îòæå, n — òåæ ïàðíå ÷èñëî. Òîìó<br />
2 2 2 2<br />
1 1<br />
22 23
Change language